Design of Fourth-Order Band-Pass Active-RC Filters Using a“Lossy” Low-pass to Band-pass Transformation

Dražen Jurišić*, Neven Mijat* and George S. Moschytz**

Abstract - In this paper the design of fourth-orderband-pass (BP) active-RC filters using a modified low-pass to band-pass (LP-BP) frequency transformation,applied to a second-order low-pass (LP) filter as aprototype, is presented. It is shown that a BP filter canbe realized by substitution of resistors and capacitors ofthe ladder in the low-pass prototype filter, by serial andparallel RC circuits in the resulting BP structure. Sucha substitution results from a so-called, “lossy” LP-BPtransformation. The design procedure is simple, and theclosed-form design equations, starting from thespecifications of a 4th-order BP filter, are presented. Acomplete step-by-step design procedure is verified on aChebyshev filter example and double-checked usingPSPICE..

1 Introduction

The design of BP filters is usually performed bymeans of the well known LP-BP frequencytransformation applied to a LP prototype filtertransfer function [1-3]. The advantage of a passive-LC filter realization lies in the existence of thecorresponding reactance LP-BP transformation,which defines the BP filter structure enabling astraightforward realization procedure, and thecalculation of the element values. The realization ofan active BP filter also usually starts with acorresponding LP prototype transfer function, butthere is no direct element transformation whichwould give a unique BP filter structure together withits component values. Instead, a designer picks aknown BP active filter structure and calculates itselements by comparing the corresponding transferfunction parameters with the parameters of thechosen structure [1,2]. In this paper we present a newprocedure for the realization of an active-RC BP filterdirectly from a given 2nd-order LP prototype, usingthe prototype impedance transformation, whichcorresponds to the so-called lossy LP-BPtransformation [3,4]. The transformation affects onlythe passive components in the circuit, while theactive elements, i. e. their number and position within

*University of Zagreb, Department of Electronic Systemsand Information Processing, Unska 3,10000 Zagreb,Croatia.E-mail: drazen.jurisic@FER.hr, neven.mijat@FER.hr,Tel: +385 1 612 94 11, Fax: +385 1 612 96 52** Swiss Federal Institute of Technology Zuerich (ETH),Signal and Information Processing Laboratory (ISI),Sternwartstrasse 7, 8092 Zürich, Switzerland. E-mail:moschytz@isi.ee.ethz.ch, Tel: +41 1 632 27 63,Fax: +41 1 632 14 46

a circuit, remain unchanged. The procedure is appliedto the realization of 4th-order BP active filters, whichcan be used as building blocks for higher-order BPfilter realizations [3,4]. Since the application ofdiscrete-component active-RC filters is generallylimited to systems in which power is at premium, wechoose a low-power filter structure using oneoperational amplifier; thus a single-amplifier second-order LP prototype [2,6] circuit is transformed into asingle-amplifier fourth-order BP filter circuit. Besidethe low power another advantage is that the resultingcircuit requires a minimum number of passivecomponents (i.e. four resistors and four capacitors) torealize the transfer function poles. Since the passiveRC-network of the filter is in the positive feedbackloop, this filter circuit belongs to the so-called class-4[1, 2] of single-amplifier active filter circuits. Thedesign procedure for the BP filter turns out to be verysimple. The closed-form design equations are given.Realizability of the filter is verified by simulationusing PSPICE.

2 Design of Fourth-Order Band-PassFilters using the LP-BP Transformation

Consider the second-order low-pass filter shown inFig. 1. The filter is known as a class-4 or Sallen andKey circuit [1,2,6].

V2

R =RG 0

R =R ( -1)F 0 βC2=C/ρ

C1=C

R1=R R2=rRA→∞

V1

Fig. 1 Second-order low-pass filter.The voltage transfer function T(s) for this circuit is

given by

22

2

012

0

1

2)(

pp

p

pLP

LPLP

LPLP

sq

s

Kasas

aKVVsT

ω+ω

+

ω=

++== , (1)

where KLP=β, 2121

01

CCRRaLPp ==ω and

1122211

2121

1 )( CRCRCCRCCRR

aq

LP

pp β−++

=ω

= (2)

and the gain GF RR /1+=β represents the gain in theclass-4 filter circuit.

The LP-BP frequency transformation is defined by

Bsss

20

2 ω+→ , (3)

where ω0 is the center frequency and B is thebandwidth of the BP filter. It doubles the filter order,and from the second-order LP filter prototype asymmetrical fourth-order BP filter with the followingvoltage transfer function is obtained:

012

23

34

22)(

asasasassbKsTBP ++++

⋅= , (4a)

where pp qBa ω=3 , 22202 2 Ba pω+ω= ,

pp qBaaa 20031 ωω== , 4

00 ω=a , 222 Bb pω= .(4b)

If the coefficients ai (i=0,1,…,3) in (4a) are known,we can calculate the LP filter prototype parametersωp, qp and LP-BP transformation parameters ω0 andB, i.e. from (4b) we have

400 a=ω ,

3

02 2a

aaq p

−= ,

Baa

p02 2−

=ω . (5)

Referring to Fig. 2a, it is assumed that the BP isgeometrically symmetrical, i.e. ωB1ωB2=ωs1ωs2=ω0

2.

Fig. 2 (a) BP filter, and (b) corresponding LPprototype specifications.

In conventional filter design the BP specificationsgiven in Fig. 2a are transformed into thecorresponding normalised LP function shown in Fig.2b. For this let

12

12

BB

sss ω−ω

ω−ω=Ω , (6)

where the frequencies ωx =2πfx [rad/s] are shown inFig. 2a. Various aids for the design of normalisedlow-pass filters are given in the literature [2]. Thecorresponding BP function can be found using (3).The center frequency ω0 and the bandwidth B of theBP filter follow from the specifications, thus:

21210 ssBB ωω=ωω=ω , 12 BBB ω−ω= . (7)In this paper we present a modified design

procedure which prewarps the pole frequencies andpole-Q factors of the normalised LP transfer function,thereby permitting a simple LP-BP transformation toprovide a simple, single amplifier fourth-orderactive-RC BP filter.

Let us assume that the LP prototype transferfunction (1) has poles s1 and s2 in the complex s-plane, as shown in Fig. 3(a).

Fig. 3 Transformation of s-variable . (a) Pole shift forδ. (b) New p-variable.

By introducing a new variable p=s+δ, where δ is areal positive constant, we apply the transformation

s=p-δ (8)to (1) and obtain a new transfer function T1(p)=T (s),with poles p1 and p2 in the complex p-plane. Thepoles of T1(p) are shifted to the right parallel to thereal axis, for an amount δ as shown in the Fig. 3 b).The new LP filter prototype transfer function is

22

2

1 )(

pp

p

p

pQ

p

KpT

Ω+Ω

+

ω= , (9a)

where

22 δ+δω

−ω=Ωp

ppp q

, δ−ω

Ω=

2/ pp

pp q

Q . (9b)

The pole-Q value of poles p1 and p2 increase whenδ increases, i.e. when the poles move closer to theimaginary axis. Since the constant δ can be freelychosen, the poles may lie even in the right-half p-plane, in which case the pole Q becomes negative,and decreases as the poles move further to the right.Note that poles in the right-half plane are permitted,since the LP-BP transformation described belowmaps the poles back into the left-half s-plane.

Application of (8) to (3), results by a “lossy”-transformation in the variable p [3], given by

δ+ω+→Bs

sp20

2

. (10)

The “lossy” LP-BP transformation (10), applied tothe transfer function T1(p) produces the same BPfilter transfer function as the conventional LP-BPtransformation (3), applied to the function T(s).

The shifted transfer function T1(p) is realized usingthe circuit in Fig. 1. Introducing impedance scalingfactors r and ρ [1], as in Fig. 1, i.e.

/ ; ; ; 2211 ρ=⋅=== CCRrRRCRR (11)and the design frequency ωd=(RC)-1, we can rewritethe transfer function (1) as follows

rRCpr

rRCp

rpTρ+⋅

β−

ρ++ρ+⋅

ρ⋅β=

)(11)()(

21 . (12)

Application of (10) to the shifted transfer functionT1(p) gives the BP transfer function (4). In order torealize this BP filter transfer function we introducethe impedance transformation which substitutes each

(b)(a)

(b)(a)

resistor of the LP prototype filter by a series RCcircuit, and each capacitor by a parallel RC circuit, asshown in Fig. 4, i. e.

aaaa

aa RsCsCR

sCRR +=⋅+→ 1

/1/1 , b

b

sCR

pC +→ 1 . (13)

CapC

Cb

R Ra

Rb

Fig. 4 RC impedance transformation.This substitution replaces the expression p⋅RC in (12)by a product of the admittance of a parallel RC circuitand the impedance of a series RC circuit, i.e.,

)/1(/1

)/1(bb

aa

aa sCRsCRsCRRCp +

⋅+→⋅ . (14)

Dividing both sides of (14) by RC, i.e. scaling by thedesign frequency of the LP prototype ωd=(RC)-1, weobtain, after some calculations,

a

b

b

a

ba

bbaa

CC

RR

CRsCRCRsp

''

''

)''/(1)''''/(12

++⋅

+→ , (15)

where /';'';';/' CCCCRRCRCRRR bbbbaaaa =⋅=⋅== (16)

The transformation (15) corresponds to the “lossy”LP-BP transformation (10). A comparison of (10)and (15) gives

bbaa CRCR ''''12

0 =ω ,ba CR

B''

1= , a

b

b

a

CC

RR

''

'' +=δ (17)

The constant δ is not entirely free. It’s minimumvalue is limited by the capacitance ratio C’b/C’a andthe resistor ratio R’a/R’b, which can be calculatedfrom (17), and they are

2

20

2

42''

BCC

a

b ω−δ±δ= , 2

20

2

42''

BRR

b

a ω−δδ= . (18)

The expression under the square root must bepositive, and the realizability constraint on δ is

B02 ω≥δ (19a)

i.e.,B

0min 2 ω=δ . (19b)

It can be shown that the sensitivity of the filteramplitude response to component tolerances isminimal for δ=δmin. For this case:

2''

'' minδ==

b

a

a

b

RR

CC

. (19c)

There is still one free parameter, and if, forexample, C’a is arbitrarily chosen, the rest of theparameters can be easily calculated. If thecapacitance ratio is denoted as c, i.e. cCC ab ='' ,then the parameters ' and ' ,' abb RRC are

cCC ab ⋅= '' , a

a CcBR

'1'⋅⋅

= , a

b CBR

'' 2

0 ⋅ω= . (20)

We conclude that the “lossy” LP-BP transformation(10), transforms resistors into series RC circuits, andcapacitors into parallel RC circuits. This procedureresults by a 4th-order BP filter shown in Fig. 5.

V2V1

R1=Ra

R3=rRa

R4= Rρ b

R2=Rb

C1=Ca

C3=C /ra

C4=C /b ρ

C2=Cb

+β

Fig. 5 Fourth-order BP filter circuit.

3 Design Examples

Example 1) As an illustration of the proposed BPfilter design procedure, we consider the filterspecifications shown in Fig. 2 which require a 4th-order BP filter for the realization. The specificationsin this case are fs1=100, fB1=150, fB2=200, fs2=300,Rs=20, Rp=0.5 (frequencies in kHz, loss in dB). Thedesign can be carried out by the following step-by-step design procedure:

i) Starting from the given filter specifications, findthe 2nd-order LP prototype filter parameters ωp, andqp: Using (6) we obtain Ωs=4, which is satisfied by a2nd-order Chebyshev LP filter with 0.5dB pass-bandripple, having the normalized pole frequencyωp=1.231342rad/s, and Q-factor qp=0.863721 [2].From (7), the center frequency is ω0=1.0883⋅106

rad/s, and the bandwidth B=3.141⋅105 rad/s, ornormalized, Bn=B/ω0=0.2887, and the coefficientsfrom (4b) are: a0=1.4027⋅1024, a1=5.3044⋅1017,a2=2.5183⋅1012, and a3=4.4787⋅105.

ii) Calculate δ such that (19) is satisfied: Wechoose the minimum value for δ, i.e. δ=2ω0/B=6.928.

iii) Calculate the new LP prototype by shifting thepoles by δ: Applying (8), the new LP prototypefunction T1(p) pole parameters are: Ωp=6.29594 andQp=-0.50648. Qp is negative, i.e., the poles lie in theright-half p-plane.

iv) Realize the new LP prototype circuitcomponents: Applying the design procedure for thefilter circuit in Fig. 1, we choose: r=1, ρ=2 andC1=C=1 [1,7]. From Ωp and Qp we calculate

1//1/)1( +ρ⋅−ρ+=β rQr p =3.39612, and the

normalized values R1= rCR p /)/(1 ρ⋅Ω= =0.22462,R2=rR=0.22462, and C2=C/ρ=0,5. Let RG=1, thenRF=RG(β-1)=2.29612.

v) Calculate the transformed impedancecomponent: Let C’a=500pF, then from (20)R’a=1837Ω; C’b=1732pF; and R’b=530.5Ω.

vi) Calculate the components of the fourth-orderBP filter: Using the normalized R and C values from

step iv), and (16) the element values are: Ra=412.8Ω;Ca=2225.9pF; Rb=530.5Ω; and Cb=1732pF. The BPfilter components are R1=Ra=412.8Ω; R2=Rb=530.5Ω;R3=rRa= 412.8Ω; R4=ρRb=1061Ω; C1=Ca=2225.9pF;C2=Cb=1732pF; C3=Ca/r=2225.9pF; C4=Cb/ρ=866pF.Note that the value of β remains the same as in the δ-shifted LP prototype. A check of the resulting filtercircuit in the example is performed using PSPICE.Fig. 6 shows the magnitude of the transfer functionα(ω)=20log TBP(jω) [dB] of the circuit in Fig. 5.

Fig. 6 Magnitude of 4th-order BP filter in example 1).Example 2) In order to analyze the influence of the

shift-constant δ and to find a possible optimal value,three different realizations of the Chebyshev BPfilter, with 0.5dB pass-band ripple (qp=0.86372,ωp=0.88602rad/s), normalized center frequency ω0=1and normalised bandwidth Bn=1, corresponding tothree values of δ, are considered. The first uses theminimal value of δ as defined in (19), the seconduses the δ-value which gives R’a⋅C’a=1/(ωp1) andR’b⋅C’b=1/(ωp2), and the third case the δ-value isarbitrarily chosen to be equal to 3. The relevantdesign parameters are presented in Table 1.

δ (R’aC’a)-1 (R’bC’b)-1 Ωp Qp β2.0 1.0 1.0 1.653 -0.556 3.272

2.13 0.69486 1.43914 1.775 -0.547 3.2923.0 0.38197 2.61803 2.590 -0.521 3.358

Table 1: Design parameters of 4th-order filter.

Fig. 7 Sensitivities of realized 4th-order BP filters.A sensitivity analysis was performed. The standard

deviation (related to the Shoeffler sensitivities) of thevariation of the logarithmic gain ∆α=8.68588∆|TBP(ω)|/|TBP(ω)|, with respect to zero mean and 1%

standard deviation of the components, was calculatedand shown in Fig. 7. As can be seen the best result isobtained for the minimal value of the shift parameterδ. Monte Carlo runs, carried out for the sameexamples, confirmed this result.

4 Conclusions

A procedure for the design of low-power fourth-order allpole BP active-RC filters is presented. Thedesign is based on a LP-BP transformation, which isapplied to a second-order LP filter prototype. Thesingle amplifier of the fourth-order BP filter providesa low output impedance and supplies positivefeedback to the passive RC-network. The latterrequires only a minimal number of components toobtain the transfer function. The initial design of thesecond-order LP active-RC filter circuit, which isused as the initial prototype, is well-known. It isshown that a “lossy” LP-BP transformationtransforms the resistors of the LP prototype circuitinto series RC combinations, and capacitors intoparallel RC combinations, resulting by a new single-amplifier 4th-order BP filter circuit. Detailed closed-form design equations for this circuit are given. Theprocedure is simple and can be extended also tosixth-and higher-(even-)order BP filters.

References

[1] G. S. Moschytz, Linear Integrated Networks:Design. New York (Bell Laboratories Series):Van Nostrad Reinhold Co., 1975.

[2] G. S. Moschytz and P. Horn, Active Filter DesignHandbook. Chichester, U.K.: Wiley 1981.

[3] N. Mijat, Low Sensitivity Structures in Realizationof Active Filters. Ph. D. Dissertation, Universityof Zagreb, Zagreb, Croatia, October, 1984.

[4] N. Mijat, G. S. Moschytz, Sensitivity ofnarrowband biquartic BP active filter block, FifthInt. Symp. on Network Theory and Design,pp.158-163, Sarajevo, Sept. 1984.

[5] N. Mijat, Realization of BP Active Filters usingUMLF structure. 29th Conference ETAN, vol. III,pp. 57-64, Nis, June 1985 (in Croatian).

[6] R. P. Sallen and E. L. Key, “A practical Methodof Designing RC Active Filters,” IRETransactions on Circuit Theory, vol. CT-2, pp.78-85, 1955.

[7] G. S. Moschytz, “Low-Sensitivity, Low-Power,Active-RC Allpole Filters Using ImpedanceTapering,” IEEE Trans. on Circuits and Systems,vol. CAS-46, No. 8, pp. 1009-1026, Aug. 1999.